**Unformatted text preview: **xi Contents
Foreword iii Preface v Chapter 1 Integers 1 Chapter 2 Fractions and Decimals 29 Chapter 3 Data Handling 57 Chapter 4 Simple Equations 77 Chapter 5 Lines and Angles 93 Chapter 6 The Triangle and its Properties 113 Chapter 7 Congruence of Triangles 133 Chapter 8 Comparing Quantities 153 Chapter 9 Rational Numbers 173 Chapter 10 Practical Geometry 193 Chapter 11 Perimeter and Area 205 Chapter 12 Algebraic Expressions 229 Chapter 13 Exponents and Powers 249 Chapter 14 Symmetry 265 Chapter 15 Visualising Solid Shapes 277 Answers 293 Brain-Teasers 311 1.1 INTRODUCTION
We have learnt about whole numbers and integers in Class VI.
We know that integers form a bigger collection of numbers
which contains whole numbers and negative numbers. What
other differences do you find between whole numbers and
integers? In this chapter, we will study more about integers,
their properties and operations. First of all, we will review and
revise what we have done about integers in our previous class. 1.2 RECALL
We know how to represent integers on a number line. Some integers are marked on the
number line given below. Can you write these marked integers in ascending order? The ascending order of
these numbers is – 5, – 1, 3. Why did we choose – 5 as the smallest number?
Some points are marked with integers on the following number line. Write these integers
in descending order. The descending order of these integers is 14, 8, 3, ...
The above number line has only a few integers filled. Write appropriate numbers at
each dot. Chapter 1 Integers 2 MATHEMATICS TRY THESE
1. A number line representing integers is given below
A B 2. C D E F –3 –2 G H I J K L M N O –3 and –2 are marked by E and F respectively. Which integers are marked by B,
D, H, J, M and O?
Arrange 7, –5, 4, 0 and – 4 in ascending order and then mark them on a number
line to check your answer. We have done addition and subtraction of integers in our previous class. Read the
following statements.
On a number line when we
(i) add a positive integer, we move to the right.
(ii) add a negative integer, we move to the left.
(iii) subtract a positive integer, we move to the left.
(iv) subtract a negative integer, we move to the right.
State whether the following statements are correct or incorrect. Correct those which
are wrong:
(i) When two positive integers are added we get a positive integer.
(ii) When two negative integers are added we get a positive integer.
(iii) When a positive integer and a negative integer are added, we always get a negative
integer.
(iv) Additive inverse of an integer 8 is (– 8) and additive inverse of (– 8) is 8.
(v) For subtraction, we add the additive inverse of the integer that is being subtracted,
to the other integer.
(vi) (–10) + 3 = 10 – 3
(vii) 8 + (–7) – (– 4) = 8 + 7 – 4
Compare your answers with the answers given below:
(i) Correct. For example:
(a) 56 + 73 = 129
(b) 113 + 82 = 195 etc.
Construct five more examples in support of this statement.
(ii) Incorrect, since (– 6) + (– 7) = – 13, which is not a positive integer. The correct
statement is: When two negative integers are added we get a negative integer.
For example,
(a) (– 56) + (– 73) = – 129 (b) (– 113) + (– 82) = – 195, etc.
Construct five more examples on your own to verify this statement. INTEGERS (iii) Incorrect, since – 9 + 16 = 7, which is not a negative integer. The correct statement is :
When one positive and one negative integers are added, we take their difference
and place the sign of the bigger integer. The bigger integer is decided by ignoring the
signs of both the integers. For example:
(a) (– 56) + (73) = 17
(b) (– 113) + 82 = – 31
(c) 16 + (– 23) = – 7
(d) 125 + (– 101) = 24
Construct five more examples for verifying this statement.
(iv) Correct. Some other examples of additive inverse are as given below:
Integer
10
–10
76
–76 Additive inverse
–10
10
–76
76 Thus, the additive inverse of any integer a is – a and additive inverse of (– a) is a.
(v) Correct. Subtraction is opposite of addition and therefore, we add the additive
inverse of the integer that is being subtracted, to the other integer. For example:
(a) 56 – 73 = 56 + additive inverse of 73 = 56 + (–73) = –17
(b) 56 – (–73) = 56 + additive inverse of (–73) = 56 + 73 = 129
(c) (–79) – 45 = (–79) + (– 45) = –124
(d) (–100) – (–172) = –100 + 172 = 72 etc.
Write atleast five such examples to verify this statement.
Thus, we find that for any two integers a and b,
a – b = a + additive inverse of b = a + (– b)
and
a – (– b) = a + additive inverse of (– b) = a + b
(vi) Incorrect, since
(–10) + 3 = –7 and 10 – 3 = 7
therefore,
(–10) + 3 ≠ 10 – 3
(vii) Incorrect, since,
8 + (–7) – (– 4) = 8 + (–7) + 4 = 1 + 4 = 5
and
8 + 7 – 4 = 15 – 4 = 11
However,
8 + (–7) – (– 4) = 8 – 7 + 4 TRY THESE
We have done various patterns with numbers in our previous class.
Can you find a pattern for each of the following? If yes, complete them:
(a) 7, 3, – 1, – 5, _____, _____, _____.
(b) – 2, – 4, – 6, – 8, _____, _____, _____.
(c) 15, 10, 5, 0, _____, _____, _____.
(d) – 11, – 8, – 5, – 2, _____, _____, _____.
Make some more such patterns and ask your friends to complete them. 3 4 MATHEMATICS EXERCISE 1.1
1. Following number line shows the temperature in degree celsius (°C) at different places
on a particular day.
Lahulspiti
–10 Shimla Srinagar
–5 0 Ooty
5 10 Bangalore
15 20 25 (a) Observe this number line and write the temperature of the places marked on it.
(b) What is the temperature difference between the hottest and the coldest places
among the above?
(c) What is the temperature difference between Lahulspiti and Srinagar?
(d) Can we say temperature of Srinagar and Shimla taken together is less than the
temperature at Shimla? Is it also less than the temperature at Srinagar?
2. In a quiz, positive marks are given for correct answers and negative marks are given
for incorrect answers. If Jack’s scores in five successive rounds were 25, – 5, – 10,
15 and 10, what was his total at the end?
3. At Srinagar temperature was – 5°C on Monday and then it dropped
by 2°C on Tuesday. What was the temperature of Srinagar on Tuesday?
On Wednesday, it rose by 4°C. What was the temperature on this
day?
4. A plane is flying at the height of 5000 m above the sea level. At a
particular point, it is exactly above a submarine floating 1200 m below
the sea level. What is the vertical distance between them?
5. Mohan deposits Rs 2,000 in his bank account and withdraws Rs 1,642
from it, the next day. If withdrawal of amount from the account is
represented by a negative integer, then how will you represent the amount
deposited? Find the balance in Mohan’s account after the withdrawal.
6. Rita goes 20 km towards east from a point A to the point B. From B,
she moves 30 km towards west along the same road. If the distance
towards east is represented by a positive integer then, how will you
represent the distance travelled towards west? By which integer will
you represent her final position from A? INTEGERS 7. In a magic square each row, column and diagonal have the same sum. Check which
of the following is a magic square.
5 –1 –4 1 –10 –5 –2 7 –4 –3 –2 0 3 –3 –6 4 –7 (i) 0 (ii) 8. Verify a – (– b) = a + b for the following values of a and b.
(i) a = 21, b = 18
(iii) a = 75, b = 84 (ii) a = 118, b = 125
(iv) a = 28, b = 11 9. Use the sign of >, < or = in the box to make the statements true.
(a) (– 8) + (– 4) (–8) – (– 4) (b) (– 3) + 7 – (19) 15 – 8 + (– 9) (c) 23 – 41 + 11 23 – 41 – 11 (d) 39 + (– 24) – (15) 36 + (– 52) – (– 36) (e) – 231 + 79 + 51
–399 + 159 + 81
10. A water tank has steps inside it. A monkey is sitting on the topmost step (i.e., the first
step). The water level is at the ninth step.
(i) He jumps 3 steps down and then jumps back 2 steps up.
In how many jumps will he reach the water level?
(ii) After drinking water, he wants to go back. For this, he
jumps 4 steps up and then jumps back 2 steps down
in every move. In how many jumps will he reach back
the top step?
(iii) If the number of steps moved down is represented by
negative integers and the number of steps moved up by
positive integers, represent his moves in part (i) and (ii)
by completing the following; (a) – 3 + 2 – ... = – 8
(b) 4 – 2 + ... = 8. In (a) the sum (– 8) represents going
down by eight steps. So, what will the sum 8 in (b)
represent? 1.3 PROPERTIES OF ADDITION AND SUBTRACTION OF INTEGERS
1.3.1 Closure under Addition
We have learnt that sum of two whole numbers is again a whole number. For example,
17 + 24 = 41 which is again a whole number. We know that, this property is known as the
closure property for addition of the whole numbers. 5 6 MATHEMATICS Let us see whether this property is true for integers or not.
Following are some pairs of integers. Observe the following table and complete it.
Statement
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii) 17 + 23 = 40
(–10) + 3 = _____
(– 75) + 18 = _____
19 + (– 25) = – 6
27 + (– 27) = _____
(– 20) + 0 = _____
(– 35) + (– 10) = _____ Observation
Result is an integer
______________
______________
Result is an integer
______________
______________
______________ What do you observe? Is the sum of two integers always an integer?
Did you find a pair of integers whose sum is not an integer?
Since addition of integers gives integers, we say integers are closed under addition.
In general, for any two integers a and b, a + b is an integer. 1.3.2 Closure under Subtraction
What happens when we subtract an integer from another integer? Can we say that their
difference is also an integer?
Observe the following table and complete it:
Statement Observation (i) 7 – 9 = – 2 Result is an integer (ii) 17 – (– 21) = _______ ______________ (iii) (– 8) – (–14) = 6 Result is an integer (iv) (– 21) – (– 10) = _______ ______________ (v) 32 – (–17) = _______ ______________ (vi) (– 18) – (– 18) = _______ ______________ (vii) (– 29) – 0 = _______ ______________ What do you observe? Is there any pair of integers whose difference is not an integer?
Can we say integers are closed under subtraction? Yes, we can see that integers are
closed under subtraction.
Thus, if a and b are two integers then a – b is also an intger. Do the whole numbers
satisfy this property? INTEGERS 1.3.3 Commutative Property
We know that 3 + 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In
other words, addition is commutative for whole numbers.
Can we say the same for integers also?
We have 5 + (– 6) = –1 and (– 6) + 5 = –1
So, 5 + (– 6) = (– 6) + 5
Are the following equal?
(i) (– 8) + (– 9) and (– 9) + (– 8)
(ii) (– 23) + 32 and 32 + (– 23) (iii) (– 45) + 0 and 0 + (– 45) Try this with five other pairs of integers. Do you find any pair of integers for which the
sums are different when the order is changed? Certainly not. Thus, we conclude that
addition is commutative for integers.
In general, for any two integers a and b, we can say
a+b=b+a
We know that subtraction is not commutative for whole numbers. Is it commutative
for integers?
Consider the integers 5 and (–3).
Is 5 – (–3) the same as (–3) –5? No, because 5 – ( –3) = 5 + 3 = 8, and (–3) – 5
= – 3 – 5 = – 8.
Take atleast five different pairs of integers and check this.
We conclude that subtraction is not commutative for integers. 1.3.4 Associative Property
Observe the following examples:
Consider the integers –3, –2 and –5.
Look at (–5) + [(–3) + (–2)] and [(–5) + (–3)] + (–2).
In the first sum (–3) and (–2) are grouped together and in the second (–5) and (–3)
are grouped together. We will check whether we get different results. (–5) + [(–3) + (–2)] [(–5) + (–3)] + (–2) 7 8 MATHEMATICS In both the cases, we get –10.
i.e.,
(–5) + [(–3) + (–2)] = [(–5) + (–2)] + (–3)
Similarly consider –3 , 1 and –7.
( –3) + [1 + (–7)] = –3 + __________ = __________
[(–3) + 1] + (–7) = –2 + __________ = __________
Is (–3) + [1 + (–7)] same as [(–3) + 1] + (–7)?
Take five more such examples. You will not find any example for which the sums are
different. This shows that addition is associative for integers.
In general for any integers a, b and c, we can say
a + (b + c) = (a + b) + c 1.3.5 Additive Identity
When we add zero to any whole number, we get the same whole number. Zero is an
additive identity for whole numbers. Is it an additive identity again for integers also?
Observe the following and fill in the blanks:
(i) (– 8) + 0 = – 8
(ii) 0 + (– 8) = – 8
(iii) (–23) + 0 = _____
(iv) 0 + (–37) = –37
(v) 0 + (–59) = _____
(vi) 0 + _____ = – 43
(vii) – 61 + _____ = – 61
(viii) _____ + 0 = _____
The above examples show that zero is an additive identity for integers.
You can verify it by adding zero to any other five integers.
In general, for any integer a
a+0=a=0+a TRY THESE
1. Write a pair of integers whose sum gives
(a) a negative integer (b) zero (c) an integer smaller than both the integers.
(e) an integer greater than both the integers. (d) an integer smaller than only one of the integers. 2. Write a pair of integers whose difference gives
(a) a negative integer.
(c) an integer smaller than both the integers.
(e) an integer greater than both the integers. (b) zero.
(d) an integer greater than only one of the integers. INTEGERS EXAMPLE 1 Write down a pair of integers whose SOLUTION (a) sum is –3 (b) difference is –5 (c) difference is 2 (d) sum is 0 (a) (–1) + (–2) = –3 or (–5) + 2 = –3 (b) (–9) – (– 4) = –5 or (–2) – 3 = –5 (c) (–7) – (–9) = 2 or 1 – (–1) = 2 (d) (–10) + 10 = 0 or 5 + (–5) = 0 Can you write more pairs in these examples? EXERCISE 1.2
1. Write down a pair of integers whose:
(a) sum is –7 (b) difference is –10 (c) sum is 0 2. (a) Write a pair of negative integers whose difference gives 8.
(b) Write a negative integer and a positive integer whose sum is –5.
(c) Write a negative integer and a positive integer whose difference is –3.
3. In a quiz, team A scored – 40, 10, 0 and team B scored 10, 0, – 40 in three
successive rounds. Which team scored more? Can we say that we can add
integers in any order?
4. Fill in the blanks to make the following statements true:
(i) (–5) + (............) = (– 8) + (............)
(ii) –53 + ............ = –53
(iii) 17 + ............ = 0
(iv) [13 + (– 12)] + (............) = ............ + [(–12) + (–7)]
(v) (– 4) + [............ + (–3)] = [............ + 15] + ............ 1.4 MULTIPLICATION OF INTEGERS We can add and subtract integers. Let us now learn how to multiply integers. 1.4.1 Multiplication of a Positive and a Negative Integer
We know that multiplication of whole numbers is repeated addition. For example,
5 + 5 + 5 = 3 × 5 = 15
Can you represent addition of integers in the same way? 9 10 MATHEMATICS TRY THESE We have from the following number line, (–5) + (–5) + (–5) = –15 Find:
4 × (– 8),
8 × (–2),
3 × (–7),
10 × (–1)
using number line. –20 –15 0 (–5) + (–5) + (–5) = 3 × (–5)
3 × (–5) = –15
(– 4) + (– 4) + (– 4) + (– 4) + (– 4) = 5 × (– 4) = –20 –16 –20
And –5 But we can also write Therefore,
Similarly –10 –12 –8 –4 0 (–3) + (–3) + (–3) + (–3) = __________ = __________ Also, (–7) + (–7) + (–7) = __________ = __________ Let us see how to find the product of a positive integer and a negative integer without
using number line.
Let us find 3 × (–5) in a different way. First find 3 × 5 and then put minus sign (–)
before the product obtained. You get –15. That is we find – (3 × 5) to get –15.
Similarly, 5 × (– 4) = – (5×4) = – 20. Find in a similar way,
4 × (– 8) = _____ = _____ 3 × (– 7) = _____ = _____
6 × (– 5) = _____ = _____ 2 × (– 9) = _____ = _____
Using this method we thus have, TRY THESE
Find:
(i) 6 × (–19)
(ii) 12 × (–32)
(iii) 7 × (–22)
We have, 10 × (– 43) = _____ – (10 × 43) = – 430
Till now we multiplied integers as (positive integer) × (negative integer).
Let us now multiply them as (negative integer) × (positive integer).
We first find –3 × 5.
To find this, observe the following pattern:
3 × 5 = 15
2 × 5 = 10 = 15 – 5
1 × 5 = 5 = 10 – 5
0×5=0=5–5 So, –1 × 5 = 0 – 5 = –5 INTEGERS –2 × 5 = –5 – 5 = –10
–3 × 5 = –10 – 5 = –15
We already have 3 × (–5) = –15 So we get (–3) × 5 = –15 = 3 × (–5) Using such patterns, we also get (–5) × 4 = –20 = 5 × (– 4)
Using patterns, find (– 4) × 8, (–3) × 7, (– 6) × 5 and (– 2) × 9
Check whether, (– 4) × 8 = 4 × (– 8), (– 3) × 7 = 3 × (–7), (– 6) × 5 = 6 × (– 5)
and (– 2) × 9 = 2 × (– 9) Using this we get, (–33) × 5 = 33 × (–5) = –165 We thus find that while multiplying a positive integer and a negative integer, we
multiply them as whole numbers and put a minus sign (–) before the product. We
thus get a negative integer. TRY THESE
1. Find: (a) 15 × (–16) (b) 21 × (–32) (c) (– 42) × 12 (d) –55 × 15 (b) (–23) × 20 = 23 × (–20) 2. Check if (a) 25 × (–21) = (–25) × 21
Write five more such examples. In general, for any two positive integers a and b we can say
a × (– b) = (– a) × b = – (a × b) 1.4.2 Multiplication of two Negative Integers
Can you find the product (–3) × (–2)?
Observe the following:
–3 × 4 = – 12
–3 × 3 = –9 = –12 – (–3)
–3 × 2 = – 6 = –9 – (–3)
–3 × 1 = –3 = – 6 – (–3)
–3 × 0 = 0 = –3 – (–3)
–3 × –1 = 0 – (–3) = 0 + 3 = 3
–3 × –2 = 3 – (–3) = 3 + 3 = 6
Do you see any pattern? Observe how the products change. 11 12 MATHEMATICS Based on this observation, complete the following:
–3 × –3 = _____ –3 × – 4 = _____
Now observe these products and fill in the blanks:
– 4 × 4 = –16 TRY THESE – 4 × 3 = –12 = –16 + 4 (i) Starting from (–5) × 4, find (–5) × (– 6)
(ii) Starting from (– 6) × 3, find (– 6) × (–7) – 4 × 2 = _____ = –12 + 4
– 4 × 1 = _____
– 4 × 0 = _____
– 4 × (–1) = _____
– 4 × (–2) = _____
– 4 × (–3) = _____
From these patterns we observe that,
(–3) × (–1) = 3 = 3 × 1
(–3) × (–2) = 6 = 3 × 2
(–3) × (–3) = 9 = 3 × 3
and (– 4) × (–1) = 4 = 4 × 1 So, (– 4) × (–2) = 4 × 2 = _____
(– 4) × (–3) = _____ = _____ So observing these products we can say that the product of two negative integers is
a positive integer. We multiply the two negative integers as whole numbers and put
the positive sign before the product.
Thus, we have (–10) × (–12) = 120 Similarly (–15) × (– 6) = 90 In general, for any two positive integers a and b,
(– a) × (– b) = a × b TRY THESE Find: (–31) × (–100), (–25) × (–72), (–83) × (–28) Game 1
(i) Take a board marked from –104 to 104 as shown in the figure.
(ii) Take a bag containing two blue and two red dice. Number of dots on the blue dice
indicate positive integers and number of dots on the red dice indicate negative integers.
(iii) Every player will place his/her counter at zero.
(iv) Each player will take out two dice at a time from the bag and throw them. INTEGERS 104
83
82
61
60
39
38
17
16
–5
–6
–27
–28
– 49
– 50
– 71
–72
– 93
– 94 103
84
81
62
59
40
37
18
15
–4
–7
–26
–29
– 48
–51
– 70
–73
– 92
– 95 102
85
80
63
58
41
36
19
14
–3
–8
–25
–30
– 47
–52
– 69
–74
– 91
– 96 101
86
79
64
57
42
35
20
13
–2
–9
–24
–31
– 46
–53
– 68
–75
– 90
– 97 100
87
78
65
56
43
34
21
12
–1
–10
–23
–32
– 45
–54
– 67
–76
– 89
– 98 99
98
88
89
77
76
66
67
55
54
44
45
33
32
22
23
11
10
0
1
–11 –12
–22 –21
–33 –34
– 44 – 43
–55 –56
– 66 – 65
–77 –78
– 88 – 87
– 99 – 100 97
90
75
68
53
46
31
24
9
2
–13
–20
–35
– 42
–57
– 64
–79
– 86
–101 96
91
74
69
52
47
30
25
8
3
–14
–19
–36
– 41
–58
– 63
–80
– 85
–102 95
92
73
70
51
48
29
26
7
4
–15
–18
–37
– 40
–59
– 62
–81
– 84
–103 94
93
72
71
50
49
28
27
6
5
–16
–17
–38
–39
– 60
– 61
–82
– 83
–104 (v) After every throw, the player has to multiply the numbers marked on the dice.
(vi) If the product is a positive integer then the player will move his counter towards
104; if the product is a negative integer then the player will move his counter
towards –104.
(vii) The player who reaches 104 first is the winner. 13 14 MATHEMATICS 1.4.3 Product of three or more Negative Integers
Euler in his book Ankitung zur We observed that the product of two negative integers is a positive integer.
Algebra(1770), was one of What will be the product of three negative integers? Four negative integers?
the first mathematicians to Let us observe the following examples:
attempt to prove
(a) (– 4) × (–3) = 12
(–1) × (...

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